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This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# New vector spaces from old

The main examples are the following:

 Given a vector space X and a subset Y of X.
Then Y is called a subspace of X if it's a vector space.
The inclusion iY: Y → X, iY(x) = x, is a linear operator.

 Given two vector spaces X and Y.
Then X×Y is the product of sets, set of all pairs (a,b) of elements in X and Y respectively.
It is a vector space with the operations
(a,b) + (a',b') = (a + a',b + b') and t(a,b) = (ta,tb).
The projections pX: X×Y → X and pY: X×Y → Y, pX(a,b) = a, pY(a,b) = b, are linear operators.

 Given a vector space X and a subspace Y. Then an equivalence relation ~ on X is defined by
x ~ y if x - y ∈ Y.
Then the quotient set X/~ is a vector space with the operation
[x] + [y] = [x + y] and q[x] = [qx].
The quotient function q: X → X/~, q(x) = [x], is a linear operator.